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\author{Class 2019 Math and Applied Math }
\title{Applied stochastic processes - Homework 08}
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%\date{2021 年 2 月 28 日}
\date{May 25, 2021}

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%\subsection{Homework 08}
%E8.1.1, E8.1.2, E8.1.4, P8.1.1, P8.1.3, E8.2.1.

\begin{document}

\maketitle

\begin{enumerate}

\item [E8.1.1.] Let $\{B(t); t \ge 0\}$ be a standard Brownian motion.
\begin{enumerate}
\item  Evaluate $\mathbb{P}\{B(4) \le 3 \mid B(0) = 1\}$.
\item  Find the number $c$ for which $\mathbb{P}\{B(9) > c \mid B(0) = 1\} = 0.10$.
\end{enumerate}

\item [E8.1.2.] Let $\{B(t);t\ge 0\}$ be a standard Brownian motion and $c>0$ a constant. 
Show that the process defined by $W(t) = cB(t/c^2)$  is a standard Brownian motion.


\item [E8.1.4.] Consider a standard Brownian motion $\{B(t);t\ge 0\}$ at times $0 < u < u+v < u+ v + w$, where $u, v, w > 0$.
\begin{enumerate}
\item  Evaluate the product moment $\mathbb{E} [ B(u)B(u+v)B(u+v+w) ]$.
\item  For $x > 0$, evaluate the product moment $$\mathbb{E} [ B(u)B(u+v)B(u+v+w)B(u+v+w+x) ].$$
\end{enumerate}


\item [P8.1.1.] Consider the simple random walk $S_n=\xi_1+\cdots + \xi_n$, $S_0=0$, in which the summands are independent with $\mathbb{P}\{\xi = \pm 1\} = 1/2$. 
\begin{enumerate}
\item  Show that the mean time for the random walk to first reach $-a < 0$ or $b > 0$ is $ab$.  
\item  Use this together with the invariance principle to show that $\mathbb{E} [T] = ab$, where 
\begin{eqnarray*}
T = T_{a,b} = \min\{t \ge 0; B(t) = -a \text{ or } B(t) = b\}, 
\end{eqnarray*}
and $B(t)$ is standard Brownian motion.
\end{enumerate}


\item [P8.1.3.] For a positive constant $\epsilon$, show that
\begin{eqnarray*}
\mathbb{P}\{  |B(t)|/t > \epsilon \} = 2[1-\Phi(\epsilon\sqrt{t})]. 
\end{eqnarray*}
\begin{enumerate}
\item  How does this behave when $t$ is large ($t \to\infty$)? 
\item  How does it behave when $t$ is small ($t \approx 0$)?
\end{enumerate}


\item [E8.2.1.] Let $\{B(t); t \ge 0\}$ be a standard Brownian motion, with $B(0) = 0$, and let 
$$M(t) = \max\{B(u); 0 \le u \le t\}.$$
\begin{enumerate}
\item  Evaluate $\mathbb{P}\{M(4) \le 2\}$. 
\item  Find the number $c$ for which $\mathbb{P}\{M(9) > c\} = 0.10$. 
\end{enumerate}



\end{enumerate}


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\end{document}

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\subsection{Homework 01}
E3.1.2, P3.1.4, E3.2.2, P3.2.4, E3.3.2, P3.3.6.

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\subsection{Homework 02}
E.3.4.1, E3.4.2, P3.4.1, P3.4.5, E3.5.1, P3.5.1. 

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\subsection{Homework 03}
E4.1.10, P4.1.1, P4.1.5, E4.3.1, E4.3.2, E4.4.2.

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\subsection{Homework 04}
E5.1.1, E5.1.7, P5.1.10, E5.2.1, P5.2.1.

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\subsection{Homework 05}
E5.3.1, E5.3.3, E5.3.7, P5.3.1, E5.4.1, E5.4.3. 

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\subsection{Homework 06}
E6.1.1, E6.1.2, P6.1.1, P6.1.2.

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\subsection{Homework 07}
E7.1.2, E7.1.3, E7.2.1, E7.2.3, P7.2.1.

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\subsection{Homework 08}
E8.1.1, E8.1.2, E8.1.4, P8.1.1, P8.1.3, E8.2.1.

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